Department of Mathematics

UNIVERSITY OF ROCHESTER

Rochester, New York 14627

Tel (585) 275-4429 or 244-9368

FAX (585) 273-4655 (at U of R)

email: RARM@math.rochester.edu

Ralph A. Raimi

Professor Emeritus

 

                                                                           25 May 2002

 

David Garbasz

12 West 72nd Street

New York, NY 10023

 

Dear Mr. Garbasz:

 

          Here is the report you requested.   I have written it as if in my own

hand, which it largely is so far as prose is concerned, though it is based on a

draft first written by Braden alone.  We have had to confer on the grades of

course.  Let me mention one curious feature of the grading. 

         

When Braden and I first evaluated state standards for the Fordham

Foundation we gave separate scores of 0 to 4 for each of our four criteria,

never thinking to average them or sum them.  The Criteria (Clarity, Content,

Reason, Negative Qualities) were of different dimensions, and we expected

the readers to consider them as such.  However, at the time of our study

there were also evaluations of standards being written by the American

Federation of Teachers and by the Council for Basic Education, each of

which was giving a summary grade of A, B, C and so on for each state. 

Chester Finn, President of the Fordham Foundation, found himself obliged

to do the same with the reports he commissioned; also, there were many

potential users of the report (State Governors, for example) who needed an

overall grade for local purposes.  Braden and I therefore summed the four

separate scores, as you will see below, and created a grading scale out of thin

air, averaging apples and oranges as it were.  The reader of this report on

Israel's Curriculum 2000 should take note of this warning, and pay more

attention to the numbers taken separately for our four criteria of judgment

than to the summed score, and more attention to the commentary than to

the numerical or alphabetical scores.

 

Part 1, Criteria and Evaluation Results

 

          The authors of this report are Lawrence Braden, a high school

mathematics teacher at St. Paul's School in Concord, New Hampshire, and

Ralph A. Raimi, Professor Emeritus of Mathematics at the University of

Rochester in Rochester, New York. The words printed here are mainly those

of Raimi, who has put them into a letter to David Garbasz, a mathematician

living in New York City.  We have been invited by Mr. Garbasz to write an

evaluation of the current draft of the Curriculum 200 (Mathematics

Curriculum), as proposed by a committee appointed by the Department for

Curricula (of Israel's Ministry of Education), to serve as a replacement for a

1998 document having similar purpose.  We were provided with an English

translation of the proposed Curriculum 2000 by Mr. Garbasz, who has said

he wants us to evaluate, for his use and for transmission to interested persons

in Israel, this Israeli "standards" document using the same criteria we had

earlier used for two such studies we made for the Thomas B. Fordham

Foundation of Washington, D.C., published in 1998 and 2000.  In those

reports we made comparative judgments of the "Standards" for school (K-12)

mathematics education as they had been published (separately) by each state

of the United States in the late 1990s at the urging of the federal government;

and we included a corresponding Japanese "standards" for comparison as

well.  These American reports can be found on the web site of the Fordham

Foundation, at <http://www.edexcellence.net/standards/math.html> and at

<http://www.edexcellence.net/library/soss2000/2000soss.html>, respectively.

 

          (As to the qualifications of the authors of these two Fordham reports,

and of our report on Curriculum 2000 contained below, we can summarize

briefly as follows: Ralph Raimi has been a professor of mathematics at the

University of Rochester since 1952, coming from having been a student and

teaching assistant in mathematics at the University of Michigan, from which

he holds degrees in physics (B.S.) and mathematics (PhD).  In earlier years

he served as a communications and radar maintenance officer in the U.S.

Army Air Forces during World War II.  Lawrence Braden has had a long

career as a teacher of mathematics in both public and private schools, in

Hawaii, Russia and New Hampshire, at elementary,  junior high school and

high school levels, and with classes of various social dimensions, from poor

students with limited English proficiency to the college-bound children of the

affluent.  In 1987 Mr. Braden was invited to Washington to receive a Presi-

dential Award for Excellence in Mathematics and Science Teaching.)

 

          The second of the Fordham reports is brief, being a revision of the earlier

reports in that it evaluated only those states which had made some

changes in the two-year interval.  It is the first of our reports, State

Mathematics Standards: An Appraisal of Math Standards in 46 States, the

District of Columbia, and Japan (1998), to which we wish to call the readers'

attention, because that volume contains a lengthy exposition of the Criteria

we used in our evaluation, criteria which are only briefly summarized in the

second (2000) report and which will also be rather briefly summarized

below.  While the full Fordham exposition of our criteria contains much

information on what we consider important in school mathematics

education, it is strictly speaking not necessary to consult it order to

understand the meaning of our evaluations given below; but it might be

useful to one who wishes to understand why we have given the weights we

did to the various aspects of the documents studied.  The present report

does not contain a full, detailed argument for the scores we awarded, and the

quotations we present below, and comment upon, are only a sampling of

what could be adduced for the purpose.  Even so, this report is considerably

more detailed than what we provided for each of the American states in 1998

and 2000, there being so many of them.

 

          In the present report, then, we will do the same for the Israeli

Curriculum 2000, hereinafter to be referred to as "Curriculum 2000" (or,

sometimes, "the document"), as we did for the American state standards,

using the identical criteria, which fall under four headings: 

 

I. Clarity refers to the success the document has in conveying to the reader

its own purpose:  What it considers important, necessary, testable, useful and

so on.  The organization of the document and the quality of its prose are

evaluated under this heading, without regard to the value of the advice

actually given, or the relevance of the curricular items themselves.  In our

Fordham reports we subdivided this heading into three subheadings,

 

          (a)  Clarity of language;

          (b)  Definiteness of the prescriptions; and

          (c)  Testability of the recommended or prescribed content.

 

We will use the same subheadings here, giving something between 0 and 4

points at each subheading and averaging the three results to obtain a maximal

score of 4 points for the entire category named Clarity, 4 being excellent or

exemplary, and 0 a bad failure.  We recognize that some of the pedagogical

recommendations of Curriculum 2000 are not "testable" in the nature of

things; this is also the case with some of the American frameworks, and

nobody is downgraded on this account.  It is the outcomes demanded, the

skills asked for, that we wish to be so plainly described that it is possible for

someone other than the teacher in the classroom to determine whether they

have been conveyed.

 

II.  Content refers to the curricular items themselves:  Does the document

ask what should be asked of students at the grade levels in question, in such

a way as to conduce to a coherent program containing the information and

developing the skills we considered optimal.  In our Fordham document we

used three subheadings, one each for primary, middle and high school levels.

In the present case, therefore, our evaluation of the Curriculum 2000 is not

quite comparable with our American states' (and the Japanese) evaluations,

as Curriculum 2000 is concerned only with the primary level and its score

will not be an average of three scores.  (In some of the Fordham evaluations

a poor "primary grades" score might have been somewhat balanced by a high

"high school" content score, or vice versa, something not possible here.)  The

reader should make proper allowances for this difference in interpreting our

overall scoring, which sums the results of the four separate criterion scores.

 

III. Reason refers to the structural organization by which the parts of

mathematics cohere as an intellectual system.  Even from the earliest grades

it is necessary that the student recognize the status of his own knowledge: 

what is empirically known, what is approximate, what is exact, what is

accepted (perhaps for the time being only) on authority, and what is

deducible from something else. Reasoning is also required in the

mathematical modeling of real phenomena, but the model is futile if the

mathematical properties of the model are not understood.  While the grade

level of Curriculum 2000 is elementary, so that formal deductive reasoning

exercises and skills cannot be expected to appear as they should in later

grades' algebra and geometry, for example, we did expect the groundwork for

such reasoning, and the appreciation of the connectedness of the subject

matter, to be inculcated from the beginning.  And indeed, those American

states whose later years' demands avoided the deductive essence of

mathematics were in fact also those whose programs for earlier years were

visibly lacking in the corresponding qualities.

 

IV. Negative Qualities refers to two common classes of failings of attitude or

philosophical stance, which we labeled

 

          (a) False doctrine; and

          (b) Inflation.

 

          By false doctrine we mean either outright mathematical error or

"advice which, if followed, will subvert instruction in the material otherwise

demanded" (or which otherwise should be demanded). Included are the

excessive reliance on calculators, excessive insistence on "real-world"

application as the criterion for value in mathematical instruction, and the

unreasonable delay (on constructivist principles) in offering direct information. 

Such prescriptions of delay, setting children adrift to guess and

argue with little direction, habituates students to avoid the learning they might

have accomplished in the same time by taking advantage of the efforts of

their more learned ancestors. 

 

          By inflation we mean "language whose only purpose is to fill paper, or

to suggest a profundity impossible for the level in question..."  Such language

weakens the respect the reader or user of the document should have for its

advice even in places where well taken. 

 

          In scoring Criterion IV we use a reverse scale, 4 points being

awarded for the absence of the quality (false doctrine or inflation) in

question, so that the grade for this criterion can be added to those of the

other three in arriving at a total.

 

          In the scoring under each criterion, "4" need not require perfection,

and "0" need not require vacuity; they are merely rankings of quality.  The

grade of A, B, C, D, or F (for "failure") is then awarded to the document as a

whole, based on a score from 0 to 16 obtained by summing the scores under

the four criterion headings.  In our Fordham Foundation reports, the grade

of A was awarded for scores of 13 or more, B for scores of 11 or more, but

less than 13, C for scores of 7 or more, but less than 11, D for scores of 4 or

more, but less than 7, and F for scores of less than 4.  We do the same here.

 

          For comparison with the score of Curriculum 2000, we should

mention that by these criteria, Japan, California, North Carolina and Ohio

received grades of A in 1998, nine states received a grade of B, seven states a

grade of C, twelve states a grade of D, sixteen states a grade of F, with four

states not judged for various reasons.  By 2000, the second Fordham report

found some improvement, and six states were awarded an A (not counting

Japan, which had not changed its document in the interim), while only eight

received an F, with two states not graded.

 

          Our grades for Curriculum 2000 are as follows:

 

I. Clarity....................................……………...… 3 points

          (a) Clarity of language ........ 4  points

          (b) Definiteness .............…. 2  points

          (c) Testability ............……. 3  points

 

II. Content ......................................………….... 1 point

 

III. Reason ...................................…………...... 1 point

 

IV. Negative qualities...........................……....... 1.5 points

          (a) False Doctrine .............. 0 points

          (b) Inflation ................…... 3 points

 

          The total score, 6.5 points out of a possible 16, rates an overall grade

of D on the Fordham Foundation scale, though not far from the lower

boundary of C.  It is worth noting that most of the document's points are the

result of good language and an avoidance, by and large, of the bureaucratic

or pretentious language that infests so many American pronouncements in

education.  It is possible that in 1998 we awarded too many points for this

part of our assessment for reasons that were rather parochial and ought not

to be applied internationally, but we are here committed to providing a grade

comparable to those we gave then.  Amusingly, one might notice that a blank

piece of paper submitted as a "Standards" or "Curriculum" would already have

four points, hence a grade of D, merely for absence of negative qualities.

 

Part 2, Commentary on the Evaluation

 

          Curriculum 2000 is a rather clearly written document of almost two

hundred pages (in our English translation), plus an introductory essay of

another 19 pages setting forth the intentions of the Planning Committee

which prepared it.  The philosophy of the Committee are certainly made

plain, and it is exemplified in what follows.  Having studied the entire

document, we have found that it would have been almost sufficient to cite

phrases of this introduction alone in support of our conclusions, and

convenient as well, since the entire Curriculum 2000 is too long to admit a

full treatment item by item here.  We did study the entire document,

however, and will submit some quotations from the main body of

Curriculum 2000 when needed.

 

          After the introduction, each of the six Grades is given a parallel

treatment:  A short essay headed An Approach to Mathematics Teaching in

the Xth Grade; a Table of Topics for each of the major divisions addressed

in each grade: Number and Operations, Geometry and Measurement, and

Data Analysis; and a section called Details and Examples.  In every grade the

document recommends to the teacher (and presumably to the textbook

writer) a ratio of 65:25:10 for the percentages of time given to the three

major divisions (Number, geometry, data).

 

          Each Table of Topics has subdivisions of several sorts in each grade,

and the items are sometimes rather summary, but then clarified further in the

third table associated with each Grade:  the Details and Examples, which are

given in two columns thus headlined.  The examples are often typical

problems to be solved, or activities to be engaged in over some period of

time, though in the document we reviewed there was some empty space

where other examples could have found a place.

 

          There is in all a good deal of overlap from one grade to another, but

it is harmless and indeed necessary if each Grade is to stand on its own, for

some readers might well consult the document for one particular grade of

interest without looking closely at the rest of the document.  However, in

reading the whole it is not always apparent where the progression of ideas is

to be found, from the simpler to the more complex, from the basics to that

which is built on the basics.  Which topics are more important than others? 

If calculators are an "inseparable part of the activities" in lessons designed to

"help the children develop a sense of self-confidence in their ability to handle

mathematics", what emphasis should be given to things that must be done in

the absence of the calculator?  The insistence on calculators begins with the

first grade:  "It is important to integrate the use of calculators throughout the

year and in daily life in parallel with developing skills of calculation orally, in

writing and in estimating."  We do not believe this, and this and similar sen-

timents are repeated in nearly identical terms in all succeeding grades.  One

would think that "self-confidence" would result from the internalizing of what

the calculator should no longer be needed for, as time goes on.

 

          On the other hand, the standard algorithms of arithmetic and the

uses of memory are downplayed when not denigrated, and much space is

given in Curriculum 2000 to explaining what need not be learned or

considered at this or that stage, as if the authors were fearful of too much

learning.  In the Introduction is written, "Studying algorithms is important to

the extent that they support the students' conceptual understanding of

numbers and their operations.  As a result, when studying topics that are

related to numbers and operations, there is no need to focus on either

calculations or the study of traditional algorithms."

 

          This statement, which we counted as a major "false doctrine", is

similar to many that are put forward in the United States, and which have led

to the utter abandonment of all arithmetic algorithms in some modern

programs that are avowedly in concord with the recommendations of the

(American) National Council of Teachers of Mathematics, but one cannot

condemn a statement on the mere grounds that its misinterpretation would

lead to evil results.  However, the statement as given in Curriculum 2000 all

but begs for such misinterpretation, for it does not say just to what extent the

traditional algorithms do or do not support the "conceptual understanding",

and it does not define the intensity of "focus", especially when the focus

warned against is on more than one thing. 

 

          As to the algorithms' "support ... [of] conceptual understanding",

Curriculum 2000 exhibits in its specific recommendations a belief that they

contribute very little, since very little in the way of algorithm is required --

indeed, they are mostly deplored -- and "student-constructed" algorithms, or

devices, are forever celebrated over the old ones besides.  And to say that

one does not "focus" on calculations and algorithms is -- at the very least -- to

say the very least.  It will not be misinterpretation that will cause followers of

Curriculum 2000 to scant calculation, both of decimal arithmetic and

fractions, to the detriment of the students' mathematical future; it will be a

correct apprehension of the import of the objectives as announced in that

document.

 

          At the 6th Grade level, the Approach section states something that is

in our view diagnostic of the entire philosophy: "The program sets goals and

limits [our emphases] for the performance of calculative algorithms in the 6th

grade ... In the division of common fractions, it is sufficient that problems are

solved at the level of insight (dividing a fraction by an integer) without

demanding the division algorithm (changing the division into multiplication

by the inverse number).  In the division of decimal numbers, the solving of

xercises based on immediate division is enough (for instance, 0.8 : 0.4)

without demanding the division algorithm..." 

 

          Yet back in Grade 5 the student was asked, albeit with fractions

whose denominators are not greater than 12, to understand "the significance

of a fraction as a quotient of division", and to handle common denominators.

 Why then the limitation?  What is the student supposed to do with

common denominators if the arithmetic of fractions is to be so limited?  It is

as if "invert and multiply" were inevitably a mindless algorithm, stunting the

student's growth, yet an important use of common denominators is one way

to make the algorithm reasonable and memorable.  It may well have been

the case that this famous algorithm was mindlessly prescribed in the bad

teaching we have all been acquainted with, but the cure for bad teaching is

not the withholding of information, information which is absolutely essential

if the student is to have any hope of mastering any part of algebra two or

three years later, where the manipulation of fractions is a notoriously difficult

pedagogical problem in the best of times.  The teaching of "invert and

multiply" is not simply the presentation of a tool the output of which is the

"answer"; it is part and parcel of the meaning of fractions, and the whole

rational number system.  Furthermore, the "self-confidence" generated by

limiting denominators to integers below 13 will soon evaporate when the

student finds he has been cheated in his earlier years, and that the principles

involved in acquaintance with fractions of all possible denominators,

principles that have been denied to him, principles which were the glory of

the development of arithmetic in Renaissance Europe, and were not

discovered by children, are more important than the decimal approximations

put out by a calculator.

 

          Furthermore, "mindlessness" of a sort is a necessary feature of

education and civilized behavior.  We do not reduce our judgments to first

principles every time we have a decision to make; we must make use of

memory, the accumulation of things we have thought about and settled in the

past, and have made into habit.  Otherwise we waste our lives in repeating

the past, painfully recapitulating both our own earlier labors and the work of

our ancestors.  It is the antithesis of education to teach children not to

remember and freely use earlier results (often called algorithms) in favor of

forever starting anew.

 

          Number and calculation is quite properly the major concern of the

early grades, with some geometry in connection with measurements such as

everyone must be able to make.  To devote 25% of the program to geometry

and measurement is not too much, if wisely done, though we consider the

10% allocated to "data analysis" excessive, since its content is so little.  Even

the average, or mean, which could be introduced as soon as division is

studied, is not demanded as an item of data analysis until Grade 5, though

students are encouraged to invent some of their own measures of dispersion

of mean, in addition to "median" and "range", which are provided by the

curriculum.  Instead, Grade 4 suggests, under the Data Analysis heading,

"Whole research projects: short surveys and projects that continue for several

lessons.  For example:  “What do the fourth graders do after school? ; Do

cars stop at the stop sign in our neighborhood? How can water be

conserved in school?" Mathematics class time is precious; these valuable

researches should be reserved for "social science", perhaps. 

 

          The geometry of Curriculum 2000 is much too primitive to deserve

their 25% even for these grade levels, for they amount mainly to exercises in

nomenclature, with the occasional warning that certain things are too difficult

to include, and sometimes the avoidance of mention of the status of some

piece of information.  A 5th grade "Investigation" begins, "It is given that the

sum of the angles of a triangle is 180 degrees.  An example for finding the

sum of the angles of a hexagon:  We divide the hexagon into triangles by ..." 

To us, and we suppose to the student or teacher as well, the status of the

remark "it is given that the sum of the angles of a triangle is 180 degrees" is

much like "Let a polygon have five sides..."  It sounds like an assumption,

which could have been otherwise.  Now it might well be that we don't expect

a Euclidean proof of this property of every triangle, but the suppression of

the status of this piece of information is certainly at odds with the ambition of

the "investigation" that follows, concerning the angles of a hexagon.

 

          Informal exercises in visualization are necessary in their place, and

can be pleasant games as well.  But a simple lesson once learned need not be

belabored for years, or even minutes, of further instruction.   One fifth grade

exercise, to discover whether it is possible for a parallelogram can have a

diagonal which has the same length as its "height", is rather clever, even if

"height" is an idea having as much to do with the placement of the

parallelogram in the ambient gravitational field as with its geometric

properties.  But this is an artificial linguistic question, a puzzle only because

of our predisposition to regard "height" as the shorter of the two possible

heights, since placing the parallelogram in the other position makes it seem a

bit unstable, gravitationally; so that textbooks seldom print pictures of "tall"

parallelograms.  The student is in this exercise trapped by a word, an artifact

of the classroom rather than of geometry; and if he discovers the common

schoolroom misconception the words have led him into he has not thereby

really learned something about parallelograms or geometry.  Such a student

can be easily taught that a parallelogram has two possible "altitudes", as a

triangle has three, and will have missed nothing by his "research" into the

possibility of the anomalous "diagonal" of this particular exercise.  This is not

to say the problem is a bad one, only that by making so much of "discovery",

as the constructivist philosophy popular in the United States does, there is

time wasted that could have been put to better use.  The idea that students

are imitating professionals in "researching" some puzzle of this sort is in fact

generally mistaken, for the first thing professionals do is to find out whether

the answer has already been found by someone else.  Real research proceeds

from a supply of previous information, not by referring to the definitions, or

the intuitions, at every stage.

 

          And while all this is going on there is real geometry that is not being

attacked.  It might be early, in Grade 6, to expect demonstrative geometry in

all its formality, but there is not even a hint in Curriculum 2000 that any such

thing is in the offing in the study of geometry.  For example, the definition of

a circle.  It is pointed out that all points of a circle (once drawn) are

equidistant from a point called the center.  Somewhere else it is noted that

there is a radius.  But if students following this curriculum were asked to

define the word "circle" they might well say it is "all points equidistant from

the center".  We have seen college students who say such things.  Yet

accuracy of definition is not beyond the power of elementary school

children, or would not be if they were exposed to careful definitions, and

required to discuss and find just where a poor definition falls short of what

they already have intuition of.  Instead, Curriculum 2000 says things like,

"Students must talk about the math they are learning... They must document

their solutions methods, their assumptions, ... other possible solutions." 

Good advice for those who have mastered the language, but in Grade 5, for

which this behavior is prescribed, such talk has proved unfruitful in many an

American classroom.  Accurate writing and speaking have to be taught, and

will not be learned by students listening to one another.  Curriculum 2000

has few if any mentions to this effect, and this failure contributes to the poor

score we have given it under the criterion called Reason. 

 

          In the Table of Topics for Grade 6, the rubric Mathematical Insight,

opposite the topics "proportion" and "finding the ratio", is the instruction,

"Non-formal experience of directly proportional situations."  There follows a

blizzard of more detailed suggestions in this connection, including

manipulatives such as "nail boards" and "strips to represent lengths" -- in

Grade 6!  But despite the details, the key words have already been written: 

"non-formal", "experience", and "situations".  The student is required not to

know too much, and to take what he does know from experience in the real

world of nail boards rather than in reasoning, for example.  In working with

proportionality one does not expect him to recite "There exists some k>0

such that y = kx" at this level, but the answer is not to consume the day, or

days, with projects illustrating proportional relationships, some of them

doubtless having to be drawn, or built, or cut out of the newspaper (which is

what happens in America these days), while a great ocean of mysteries

concerning fractions lies all before him.  Why not give him a little help?

 

          The students are asked somewhere in Grade 6 to obtain decimal

equivalents to such fractions as 1/2 and 3/4, but the mental strain of sys-

tematically finding (even for simple examples) the "whole" when a given

percentage is known is considered beyond their powers:  "Finding the whole

quantity from a given percentage (not necessarily by building an exercise of

calculation)" is the instruction as it appears in the translation we have.  Why

not a calculation?  If 3/5 (or 60%) of a number N is given surely N is 5/3 the

given number. And even if it is a complicated percentage such as 61% (i.e.

61/100) of N that is known, then N is as surely 100/61 times the known

number.  Calculators, by the way, have nothing to do with this understanding,

which ought to be commonplace even in the 5th grade, else what are

fractions for?  And surely by the 6th grade one need no longer buy toys to

illustrate fractions, even if so urged by publishing houses.

 

          In another place (5th Grade Table of Topics for geometry) it is

bravely stated that the student should know the difference between a

statement and its converse, but the entire Curriculum 2000 contains nothing

resembling a theorem.  Again, here the advertising is excessive, and

constitutes inflation by the rubrics of our evaluation.  Theorems do abound

at the elementary level, though they are not usually construed as such.  Every

algorithm is a theorem, stating that a certain definite procedure will infallibly

yield a certain definite result.  The intellectual value of these algorithms and

the analysis that shows them to be correct far outweigh the supposed benefits

of the use of the algorithms in daily life.  Every adult will of course use a

calculator when faced with an intricate calculation of practical use, but it does

not follow that children should be taught to do the same, especially with

simple calculations such as appear in Grades 1 and 2, where Curriculum

2000 prescribes calculators as an inseparable part of the curriculum.  There

are many things adults do in daily life that we do not prescribe for children,

for the process of learning is an artificial one, not to be confused with the

experiences of daily life (though of course the applications should also be

taught when possible). 

 

          Some things really have nothing to do with "experience" and

"situations", and are not even understandable when only given "informally",

but are stepping-stones to other things in real life.  Arpeggios are not real

music, but are essential in learning real music.  For ages in America children

have built vocabulary and reading ability with the aid of "spelling bees", than

which nothing could be more artificial.  Knowing the algorithm by which the

fraction 11111/13 can be converted to a decimal appears impractical (our

calculator gives the answer "854.69230770", from which one would be hard

put to deduce any periodicity of the digits in the long run); yet the fact that

any such quotient is ultimately periodic is something students in the high

schools will be asked to understand -- unless the high school mathematics

curriculum is by then similarly disfigured by the insistence on calculators as a

path to "understanding".  It is the apparently useless algorithm that in the long

run will yield up a periodicity; and it is furthermore a theorem that, conversely,

a periodic infinite decimal expansion represents a fraction.  Both theorems

are fairly profound, and not suited to the 6th grade curriculum, but the understanding

and skill at elementary calculation must be achieved by

that stage if the later theorems are ever to be accessible.  It is not fair to

children to cut off their later development by building a false self-confidence

based on failure to offer challenges.

 

          One more example, where Curriculum 2000 deliberately scants the

prefiguring of important later developments, occurs in the discussions of area

and volume.  Volume is rather informally understood as something

measurable by the pouring of liquids, and area as something to be derived

when possible from tilings, though it is hard to trace the development of the

idea of area in Curriculum 2000, apart from the fact that finite partitions of

polygons preserve the original area when reassembled.  (Much is made of

the tangram, for some reason.)  At any rate, here are extracts from the 6th

grade Details and Examples for geometry, concerning properties of circles:

 

          "The connection between the diameter and the circumference of a

circle:  This investigative activity can be performed in various ways.  Its aim is

to reach the following conclusions: 

          One.  There is a connection between the diameter of a circle and its

circumference.  This connection exists in every circle, large and small. 

          Two.  The ratio of the diameter to the circumference is equal to

nearly 3.  There is no need to reach the exact formula."

 

          "There are various ways of measuring the approximate area of a circle

(the method of measuring area by units of tiles is not suitable for a circle).”

 

          "The area of a circle equals approximately the area of three squares

whose sides are equal to the radius of the circle.  There is no need to reach

the exact formula."

 

          In its anxiety to forestall the mere "memorization" of formulas, the

program prescribes "investigative activities" resulting in formulas guaranteed

to be informal and incorrect, all the while warning against "exact" formulas. 

There is neither practical value nor intellectual value in such uses of time. 

But time-wasting is not the only evil in this passage, since "the method of

measuring area by units of tiles" is not a mere measuring device; it is the very

definition of area.  Its importance is enormous, and in principle it is the only

method of measuring the area of a circle.  It would be better to leave off all

these geometric activities and adventures in favor of some straightforward

information, with verification and experience as needed over time.  The

same must be said, even more forcefully, about the arithmetic.

 

          The sections on Data Analysis fortunately are scheduled to take up

only 10% of instructional time, for they are quite empty of content.  Students

learn what the median is, and can spend a lot of time collecting numbers and

finding the median and range.  When they are old enough to add some

numbers and divide (with a calculator) by the size of the sample, they are

permitted to add "average" to their vocabulary.  It is not possible to see what

else there is in this 10% of the first six years of mathematics prescribed by

Curriculum 2000, except for some graphical displays and "projects", with the

intellectual content no more than can be accomplished in a week or two in

middle school or high school, where it can be combined with some

probability considerations.

 

          All told, then, we give Curriculum 2000 a more or less failing grade. 

We should mention here that our instructions were to grade Curriculum

2000 according to the criteria we used in our two Fordham reports.  This

means that large parts of Curriculum 2000 have had to be ignored, since they

refer to pedagogy alone, something that forms part of a good number of

American states' "frameworks", which Curriculum 2000 resembles more than

the bare-bones "Standards" which generally form only part of a framework in

states that publish in this form.  Yet the pedagogical advice is not devoid of

information concerning the intent of the standards part of the document, for

it permits one to deduce something of the motivation of the authors.

 

          Motivation is always dangerous to attempt to fathom, especially the

motivation behind something one disapproves of.  In the case of Curriculum

2000, however, some of the motivation is expressly stated, and so it is safe to

mention it. In the Introduction it is written that "The committee once again

stresses that the main goal of the study of mathematics in the elementary

school is to avoid failure, and build each young student's confidence that he

or she can study math with enjoyment and success."

 

          It is possible that the translation into English is imperfect here, for

surely avoidance of failure and the building of confidence cannot be the

main goal, but even with modification this statement is an announcement of

values we see richly exemplified in the sequel.  The American NCTM

(National Council of Teachers of Mathematics) has put forward such a

position beginning in 1980, culminating in its Standards of 1989 which have

had a great impact on American schools, and evidently abroad as well.  The

NCTM has retreated somewhat, though only somewhat, in its second edition

of the Standards, now called PSSM (Principles and Standards for School

Mathematics), published in 2000.  Meanwhile, in America, there has been a

considerable reaction to the "dumbing down" of the curriculum, whose

purpose is the avoidance of the announcement of failure and the building of

self-esteem, and that reaction has now garnered several years of experience

in California, where an avowedly anti-NCTM framework has been adopted,

and corresponding textbook selections, and the results have been quite the

opposite of the predictions of widespread failure and despair. 

 

          Much depends on the quality of teaching, of course, and there are

environmental factors (common to many other countries as well) that are still

insuperable whatever the curriculum might be, at least for some students; but

the removal of content from the curriculum is only a hiding of failure.  One

does not "avoid failure" by simply avoiding all challenge that could possible

lead to failure; one merely avoids its announcement. And one does not really

build self-confidence with constant praise; a student soon learns that there

has to be something to be confident about, and indeed comes to suspect all

praise, even if deserved, since incessant praise and avoidance of "judgment"

have given him no standard of value.

 

          We see Curriculum 2000 as a long step in the direction that our own

NCTM took twenty years ago, with no success.  America has not risen in

international comparisons in mathematics, its students have still been failing

(though sometimes the failure is concealed under other titles) in the usual

numbers, and its "successful" students, who are the future of its technology

and its general intellectual life, are coming to college with less and less

necessary background all the time.  Our technical industries are

consequently becoming the employers of immigrants from the Far East,

Russia and India, where early mathematical education is the antithesis of

what NCTM promulgates.  And it is not only technology, or science, or

mathematics, that is being badly served by this conversion of mathematics

instruction into happy games, but the whole of our culture, of which

mathematics is an inseparable part.

 

          It is not possible for professional mathematicians to "take over" the

teaching of mathematics in the schools.  If they could, or did, they would

cease to be mathematicians; that is not their trade.  Some few have taken an

interest in the matter, but their advice is generally rebuffed by those in the

profession of teaching mathematics or training the teachers of school

mathematics, professors in teachers colleges and so on, on the grounds that

as mathematicians they don't understand the nature of pedagogy or its

problems.  And it is probable that the average mathematician would not be

successful if drafted to teach at the 5th grade level, or at least would not be

comfortable there, and might therefore not work with the proper enthusiasm.

  But it does not follow that he does not know what can be learned at that

level, and ought to; his experience as a mathematician will give him an insight

into the ordering of topics, the distinctions between relevant and irrelevant

subject matter, the purpose of sometimes obscure relationships and

notations, that are not fully appreciated by those proposing curricula and

examinations for the schools.

 

          If we can make any single proposal to a committee seeking to write a

mathematics curriculum, in any country in the world today, including the

United States, and whether at elementary or advanced level, it would have to

be this:  Make sure the next round of writing is conducted with the active

participation (and not just editing or after-the-fact criticism) of as many

mathematicians as of professional school educators. 

 

          This recommendation is not only the recommendation of Ralph Raimi,

the mathematician, but equally that of Lawrence Braden, the teacher.

It is plainly not the advice the Ministry of Education in Israel has received

when preparing Curriculum 2000.

 

                                                Sincerely yours,

 

                                                Ralph A. Raimi

                                                Lawrence Braden